A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Stochastic continuity

Let $(X_t)_{t \in \mathbb{R}}$ be a square-integrable real-valued process with a continuous mean value function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ and a continuous covariance function ...
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8 views

Applying the optional sampling theorem

I have a Poisson process $N(t)$ with parameter $\lambda$, i.e., $N(t)$ has a Poisson distribution with parameter $\lambda t$. I know that $N(t)-\lambda t$ (and its stopped version) is a martingale and ...
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14 views

Differential equation whose solution is Erlang distribution

I am working on a proof (Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)) and got stuck. Now I would like try ...
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Large Deviation Theory

Consider a differential equation of the form: $$dX_0 = f(X_\epsilon) dt$$ and it's perturbed form: $$dX_\epsilon = f(X_\epsilon) dt+ \epsilon dW(t)$$ It's well-known that if one assumes $f$ is ...
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Existence of the Brownian Motion using the Kolmogorov extension theorem

Kolmogorov extension theorem: Let $T$ denote some interval (thought of as "time"), and let $n \in \mathbb{N}.$ For each $k \in \mathbb{N}$ and finite sequence of times $t_{1}, \dots, t_{k} \in T$, ...
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Reflection principle application

I want to calculate the probability: \begin{equation*} P(W_4>2, \inf_{0\leq t\leq4} W_t >-1) \end{equation*} and $W$ is a Wiener process. I tried: \begin{equation*} P(W_4>2, \inf_{0\leq ...
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1answer
23 views

Applying Picard-Lindelöf iteration to a stochastic integral equation

Suppose we have the following stochastic integral equation (we can make it an SDE) where $W$ is a standard Brownian motion $$ X_t = 1 + \int_0^t X_s d W_s. $$ I want to show that by using Picard ...
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1answer
42 views

Application of Ito's formula

I have the following process: \begin{equation*} X_t= \exp \left(\int_{0}^{t}s \, dB_s-\frac{t^3}{6} \right), \end{equation*} where $B$ is a Browinan motion. My textbook asks to write Ito's formula ...
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1answer
21 views

Brownian motion proof of Dirichlet problem

I am reading the proof of the Dirichlet theorem stated in the following form: Theorem: Let $D$ be a bounded domain in $\mathbb{R}^d$ such that every boundary point satisfies the Poincare cone ...
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26 views

If $B$ is a Brownian motion and $B'_t:=B_{T+t}-B_T$ for a fixed $T$, then $(B'_t,t\ge 0)$ and $(B_s,0\le s\le T)$ are independent

Let $B=(B_t,t\ge 0)$ be a Brownian motion and $$B'_t:=B_{T+t}-B_T\;\;\;\text{for }t\ge 0$$ for some $T\ge 0$. Especially, $B$ has independent increments, i.e. ...
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21 views

Poisson process. Patients arriving te the ER.

People arrive to the ER of a hospital following a poisson process with $\lambda=2.1$ patients/hour. One of each 28 who arrives under this condition, dies. Calculate the probability of: (a) At least ...
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13 views

Martingale property-Girsanov Theorem. [on hold]

Suppose $\mathbb{Q} \ll \mathbb{P}|_{\mathcal{F}_t}$ with $\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\mathcal{F}_T}=Z_T$ . Then $Z_t:=\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\mathcal{F}_t}$ is a ...
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29 views

Convergence of Ornstein-Uhlenbeck process

Let $W$ be a standard Brownian motion. Let $\alpha,\sigma^2 >0$, and let $X_0$ be a $\mathbb{R}$-valued random variable with distibution $\nu$ that is independent of $\sigma(W_t,t\geq 0)$. Now ...
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1answer
10 views

Showing that a set is included in a filtration at a stopping time

The title may sound strange. Sorry for that but the question is short and easy to understand. I have a set $A \in \mathcal{F}_t$ where $(\mathcal{F}_t)$ is a filtration on some probability space. For ...
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13 views

Obtaining the transition probability matrix

Seven black balls are distributed among two persons $A$ and $B$ having urns $X_A $ and $X_B$ with three balls in $X_A$ and four in $X_B$. One white ball is in either $X_A $ or $X_B$. A game consists ...
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1answer
21 views

Cadlag process and measurability.

Let $(\Omega,(\mathcal{F_t})_{t\geq0},P)$ be a filtered probability space and $X=(X_t)_{t\geq0}$ a real-valued adapted cadlag process. Let $A\subset\Omega$ (resp. $B\subset\Omega$) be the event that ...
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1answer
9 views

Ito's formula for this stochastic differential - please explain this step?

Referring to those two lines, can someone please explain how those results were obtained? My understanding is, the following formula is being referenced: $$dV_t = dV(S_t,t) = \frac{\partial ...
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8 views

Finite-Difference Scheme for a Non-Linear PDE?

I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? ...
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1answer
26 views

Density of a compound poisson process.

People arrive to a bank according to a possion process $N(t)$ with $\lambda = 1$ client/minute. Each client makes a deposit $Y \sim \mathrm{Unif}\{1,2\}$ in thousand dollars. Calculate the probability ...
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19 views

Conditioning a Brownian Bridge

Brownian Bridge: Consider the standard Brownian $X(t)$ motion conditioned to land at $b$ at time $1$. This means that for every trajectory {$(t,X(t)), 0 \leq t \leq 1$} of this process, its initial ...
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1answer
29 views

Itô integral of an elementary process

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t,t\ge 0)$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t,t\ge 0)$ be a stochastic process on ...
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Can we derive the PDE followed by a marginal transition probability density?

A pair of correlated stochastic processes follow the SDEs \begin{align} dX_t&=a(t,X_t)\,b(t,Y_t)\,dt+c(t,X_t)\,d(t,Y_t)\,dW_t, &&X_0=\bar{x}\\ dY_t&=f(t,Y_t)\,dt+g(t,Y_t)\,dZ_t, ...
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1answer
17 views

Showing that a Markov jump process is a Feller-Dynkin process

Let $E$ be a countable state space with $\sigma$-algebra $2^E$ and $X_t$ a Markov jump process with transition function $$P_t(x,y) = \sum_{n=0}^\infty e^{-\lambda t}\frac{(\lambda ...
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2answers
51 views

Poisson Process. Expected time of three fishermen catching at least three fish.

Three fishermen are fishing, we model the fishing as a Poisson Process of rate $2.5$ fish/hour. The fishermen leave only when each of them them has caught at least 3 fish, we call this leaving time ...
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22 views

A Quick Question on Filtrations and Stopping Times

Let $(\Omega,\mathcal{F},P)$ be a probability space with filtration $\{\mathcal{F}_t:t\geq0\}$. Define $\mathcal{F}_{t+}=\bigcap\limits_{s>t}\mathcal{F}_s$. If $\tau$ is a random time and $\forall ...
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38 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
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1answer
38 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even tough its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
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44 views
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Does the random variable $f(\tau)M_\tau$, where $M$ is a martingale and $\tau$ is a stopping time, have zero expectation?

Suppose that $M:=\{M_t\}_{t\geq0}$ is a martingale adapted to some filtration $\mathcal{F}:=\{\mathcal{F}_t\}_{t\geq0}$ with $M_0\equiv0$ and that $\tau$ is an $\mathcal{F}_t$-stopping time. Suppose ...
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1answer
29 views

Existence of steady state distribution for finite state Markov chains

Let's assume a Markov chain has 2 recurrent classes and a transient state from which we can go to either of the recurrent classes. If one of those recurrent classes is periodic, would it effect the ...
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1answer
39 views

What is the distribution of a stochastic process?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be a random variable on ...
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Question about Weiner process [on hold]

Suppose $X(t)$ is a Weiner process. How would I be able to obtain the distribution of $\sup_t (X(t)-2t)$?
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21 views

Scaling Gaussian process

If $X\sim N(0,I)$, then $VX \sim N(0,VV^\top)$. Consider the Gaussian process $Y\sim \mathcal{G}(0,k(t,t')),t,t'\in[0,1]$, that is any finite-dimensional distribution of $Y$ is a multivariate normal. ...
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21 views

Why is the expectation of essential supremum equal the supremum of expectations

Let $\{X_i\}$ be a sequence sequence of nonnegative r.v. which has the lattice property. This implies that there is a sequence of indices $\{i_n\}$ so that $\{X_{i_n}\}$ is nondecreasing and ...
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1answer
18 views

Can an indicator function be a valid Radon Nikodym derivative?

Take a process $X_t$ defined on a canonical space with probability $\mathbb{P}$. Can the indicator function $\mathbb{1}_{X_t< U}$ be a Radon Nikodym derivative? That is can we have a measure ...
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1answer
52 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
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27 views

how to prove this function is a probability measure in $U_B$

Let $(\Omega, U, P)$ be a probability space. and $B\in U$, $P(B)\gt 0$ $U_B =\{A: A=B\cap C, C\in U\}$ its class in $\Omega$ is a $\sigma$-algebra and $P_B : U_B \to \Bbb R$ $A \to ...
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22 views

Transition probability matrix

In the article here it had this question. A walker moves on two positions a and b. She begins at a at time 0, and is at a next time as well. Subsequently, if she is at the same position for two ...
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1answer
48 views

Long term probability in Markov Chains

I was practicing some questions on transition probability matrices and I came up with this question. You have 3 coins: A (Heads probability 0.2),B (Heads probability 0.4), C (Heads probability ...
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22 views

Preparation for research in statistical inference for stochastic processes [on hold]

I am interested in building capacity to do research (and ultimately building a career) in statistical inference for continuous time stochastic processes. I am seeking advice on how best to go about ...
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12 views

Ross Intro to Probability Models--Example 4.4

Can someone explain to me please how we derived the Transition Matrix? Why we decided to put $P_{00} =0.7$ and $1 - P_{00} = P_{02}$. I just don't see it the way Ross defined the different states. ...
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1answer
43 views

Separable generating function of a pair of dependent discrete random variables

Independence is sufficient but not necessary for the generating function of the sum of two random variables to be the product of their individual generating functions. I am trying to come up with an ...
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1answer
25 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...
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1answer
36 views

Continuity of $x \mapsto E_{x}[F]$, Brownian motion

I have a question about Brownian motion. Let $(\Omega,\mathcal{F},P)$ be a Probability space and $(B_{t})_{t \in [0,\infty[}$ be a standard $1$-dimensional Brownian motion defined on ...
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1answer
37 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
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10 views

Cholesky decomposition of positive semi-definite covariance matrix

I am trying to simulate a column vector of random complex variables, $\boldsymbol{x}$, which has a has a given covariance matrix: $$ \boldsymbol{C}=E\left[\boldsymbol{x}\boldsymbol{x}^{*}\right] $$ ...
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27 views

what is determinantal process?

Would anyone please explain what does this mean? A random point process $P$ on a discrete base set $Y = \{1,\ldots,N\}$ is a probability measure on the set $2^Y$ of all subsets of $Y$. Let $K$ ...
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1answer
37 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
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76 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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1answer
32 views

Strong Markov Property and Product of Expectations

Let $(B_{t})_{t\geq0}$ be a Brownian motion and let $\tau=\inf\left\{ t\geq0:B_{t}\leq-4\right\} $ be a stopping time. Then the strong Markov property ensures that e.g. $A:=\left\{ ...
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1answer
51 views

What is the integral of a family of diffusion processes? [on hold]

Let $S$ be an infinite subset of $[0,1]$. For all $s \in S$, let W_s(t) be a standard Wiener process. Definite P(s)_t = \mu(P,s,t) dt + \sigma(P,s,t) dW^s_t Can we characterize? $$F_t= \int_S P(s)_t ...